This is a first graduate course on advanced simulation. Topics include: Basic simulation methods (inversion, rejection, Box Muller, ratio of uniforms), Monte Carlo Integration, variance reduction methods (importance sampling, antithetic variates, control variates, multilevel Monte Carlo), unbiased simulation, Markov chain Monte Carlo, Metropolis-Hastings, Gibbs Sampler, Non-Reversible Markov chain methods, Unbiased Markov chain Methods, Particle Filters, Sequential Monte Carlo samplers, Particle Markov chain Monte Carlo, Particle Gibbs samplers, Coupled-Conditional Particle Filters, Advanced multilevel methods, applications to stochastic differential equations and Bayesian inverse problems.